Loading doc/usermanual/05_models/05_impact_ionization.md +40 −41 Original line number Diff line number Diff line Loading @@ -8,16 +8,16 @@ weight: 5 Allpix Squared implements charge multiplication via impact ionization models. These models are only used by propagation modules which perform a step-by-step simulation of the charge carrier motion. The gain $g$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $`\alpha`$ and the length of the step $l$ performed in the respective electric field. If the electric field strength stays below a configurable threshold $`E_{\textrm{thr}}`$, unity gain is assumed: The gain $`g`$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $`\alpha`$ and the length of the step $`l`$ performed in the respective electric field. If the electric field strength stays below a configurable threshold $`E_{\text{thr}}`$, unity gain is assumed: ```math \begin{equation} g (E, T) = \left\{ \begin{array}{ll} e^{l \cdot \alpha(E, T)} & E > E_{\textrm{thr}}\\ 1.0 & E < E_{\textrm{thr}} e^{l \cdot \alpha(E, T)} & E > E_{\text{thr}}\\ 1.0 & E < E_{\text{thr}} \end{array} \right. \end{equation} Loading @@ -27,7 +27,7 @@ The the following impact ionization models are available: ## Massey Model The Massey model \[[@massey]\] describes impact ionization as a function of the electric field $E$. The Massey model \[[@massey]\] describes impact ionization as a function of the electric field $`E`$. The ionization coefficients are parametrized as ```math Loading Loading @@ -84,8 +84,8 @@ For holes, two sets of impact ionization parameters $`p = \left\{ a_{\infty}, b \begin{equation} p = \left\{ \begin{array}{ll} p_{\textrm{low}} & E < E_{0}\\ p_{\textrm{high}} & E > E_{0} p_{\text{low}} & E < E_{0}\\ p_{\text{high}} & E > E_{0} \end{array} \right. \end{equation} Loading @@ -96,28 +96,27 @@ Sentaurus TCAD user manual as: ```math \begin{equation} \label{eq:multi:man:gamma} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{eV/K} \cdot T} \right)^{-1} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1} \end{equation} ``` The value of the reference temperature $T_0$ is not entirely clear as it is never stated explicitly, a value of The value of the reference temperature $`T_0`$ is not entirely clear as it is never stated explicitly, a value of $`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in Allpix Squared are taken from the abstract of \[[@overstraeten]\] as: ```math \begin{equation*} \begin{split} E_0 &= 4.0\times 10^{5} \,\textrm{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\textrm{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\textrm{V/cm}\\ E_0 &= 4.0\times 10^{5} \,\text{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\text{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\text{V/cm}\\ \end{split} \qquad \begin{split} a_{\infty, h, \textrm{low}} &= 1.582\times 10^{6} \,\textrm{/cm}\\ a_{\infty, h, \textrm{high}} &= 6.71\times 10^{5} \,\textrm{/cm}\\ b_{h, \textrm{low}} &= 2.036\times 10^{6} \,\textrm{V/cm}\\ b_{h, \textrm{high}} &= 1.693\times 10^{6} \,\textrm{V/cm}\\ a_{\infty, h, \text{low}} &= 1.582\times 10^{6} \,\text{/cm}\\ a_{\infty, h, \text{high}} &= 6.71\times 10^{5} \,\text{/cm}\\ b_{h, \text{low}} &= 2.036\times 10^{6} \,\text{V/cm}\\ b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm}\\ \end{split} \end{equation*} ``` Loading @@ -135,7 +134,7 @@ features a linear dependence on the electric field strength $`E`$. The coefficie \end{equation} ``` The two parameters $a, b$ are temperature dependent and scale with respect to the reference temperature The two parameters $`a, b`$ are temperature dependent and scale with respect to the reference temperature $`T_0 = 300 \,\text{K}`$ as: ```math Loading @@ -152,16 +151,16 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ ```math \begin{equation*} \begin{split} a_{300, e} &= 0.426}{/V}\\ a_{300, e} &= 0.426 \,\text{/V}\\ c_{e} &= 3.05\times 10^{-4}\\ b_{300, e} &= 4.81\times 10^{5} \,\textrm{V/cm}\\ b_{300, e} &= 4.81\times 10^{5} \,\text{V/cm}\\ d_{e} &= 6.86\times 10^{-4}\\ \end{split} \qquad \begin{split} a_{300, h} &= 0.243 \,\textrm{/cm}\\ a_{300, h} &= 0.243 \,\text{/cm}\\ c_{h} &= 5.35\times 10^{-4}\\ b_{300, h} &= 6.53\times 10^{5} \,\textrm{V/cm}\\ b_{300, h} &= 6.53\times 10^{5} \,\text{V/cm}\\ d_{h} &= 5.67\times 10^{-4}\\ \end{split} \end{equation*} Loading @@ -172,7 +171,7 @@ This model can be selected in the configuration file via the parameter `multipli ## Bologna Model The Bologna model \[[@bologna]\] describes impact ionization for experimental data in an electric field range from $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \,\text{\celsius}`$. The impact ionization $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \,^{\circ}\text{C}`$. The impact ionization coefficient takes a different form than the previous models and is given by ```math Loading Loading @@ -200,33 +199,33 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ ```math \begin{equation*} \begin{split} a_{0, e} &= 4.3383}{V}\\ a_{1, e} &= -2.42e-12}{V}\\ a_{0, e} &= 4.3383 \,\text{V}\\ a_{1, e} &= -2.42\times 10^{-12} \,\text{V}\\ a_{2, e} &= 4.1233\\ b_{0, e} &= 0.235 \,\textrm{V}\\ b_{0, e} &= 0.235 \,\text{V}\\ b_{1, e} &= 0\\ c_{0, e} &= 1.6831e4}{V/cm}\\ c_{1, e} &= 4.3796}{V/cm}\\ c_{0, e} &= 1.6831\times 10^{4} \,\text{V/cm}\\ c_{1, e} &= 4.3796 \,\text{V/cm}\\ c_{2, e} &= 1\\ c_{3, e} &= 0.13005}{V/cm}\\ d_{0, e} &= 1.2337e6}{V/cm}\\ d_{1, e} &= 1.2039e3}{V/cm}\\ d_{2, e} &= 0.56703}{V/cm}\\ c_{3, e} &= 0.13005 \,\text{V/cm}\\ d_{0, e} &= 1.2337\times 10^{6} \,\text{V/cm}\\ d_{1, e} &= 1.2039\times 10^{3} \,\text{V/cm}\\ d_{2, e} &= 0.56703 \,\text{V/cm}\\ \end{split} \qquad \begin{split} a_{0, h} &= 2.376 \,\textrm{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\textrm{V}\\ a_{0, h} &= 2.376 \,\text{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\text{V}\\ a_{2, h} &= 1\\ b_{0, h} &= 0.17714 \,\textrm{V}\\ b_{1, h} &= -2.178\times 10^{-3} \,\textrm{/K}\\ b_{0, h} &= 0.17714 \,\text{V}\\ b_{1, h} &= -2.178\times 10^{-3} \,\text{/K}\\ c_{1, h} &= 0\\ c_{1, h} &= 9.47\times 10^{-3} \,\textrm{V/cm}\\ c_{1, h} &= 9.47\times 10^{-3} \,\text{V/cm}\\ c_{2, h} &= 2.4924\\ c_{3, h} &= 0\\ d_{0, h} &= 1.4043\times 10^{6} \,\textrm{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\textrm{V/cm}\\ d_{2, h} &= 1.4829 \,\textrm{V/cm}\\ d_{0, h} &= 1.4043\times 10^{6} \,\text{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\text{V/cm}\\ d_{2, h} &= 1.4829 \,\text{V/cm}\\ \end{split} \end{equation*} ``` Loading Loading
doc/usermanual/05_models/05_impact_ionization.md +40 −41 Original line number Diff line number Diff line Loading @@ -8,16 +8,16 @@ weight: 5 Allpix Squared implements charge multiplication via impact ionization models. These models are only used by propagation modules which perform a step-by-step simulation of the charge carrier motion. The gain $g$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $`\alpha`$ and the length of the step $l$ performed in the respective electric field. If the electric field strength stays below a configurable threshold $`E_{\textrm{thr}}`$, unity gain is assumed: The gain $`g`$ is calculated for all models as exponential of the model-dependent impact ionization coefficient $`\alpha`$ and the length of the step $`l`$ performed in the respective electric field. If the electric field strength stays below a configurable threshold $`E_{\text{thr}}`$, unity gain is assumed: ```math \begin{equation} g (E, T) = \left\{ \begin{array}{ll} e^{l \cdot \alpha(E, T)} & E > E_{\textrm{thr}}\\ 1.0 & E < E_{\textrm{thr}} e^{l \cdot \alpha(E, T)} & E > E_{\text{thr}}\\ 1.0 & E < E_{\text{thr}} \end{array} \right. \end{equation} Loading @@ -27,7 +27,7 @@ The the following impact ionization models are available: ## Massey Model The Massey model \[[@massey]\] describes impact ionization as a function of the electric field $E$. The Massey model \[[@massey]\] describes impact ionization as a function of the electric field $`E`$. The ionization coefficients are parametrized as ```math Loading Loading @@ -84,8 +84,8 @@ For holes, two sets of impact ionization parameters $`p = \left\{ a_{\infty}, b \begin{equation} p = \left\{ \begin{array}{ll} p_{\textrm{low}} & E < E_{0}\\ p_{\textrm{high}} & E > E_{0} p_{\text{low}} & E < E_{0}\\ p_{\text{high}} & E > E_{0} \end{array} \right. \end{equation} Loading @@ -96,28 +96,27 @@ Sentaurus TCAD user manual as: ```math \begin{equation} \label{eq:multi:man:gamma} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\textrm{eV}}{2 8.6173\times 10^{-5} \,\textrm{eV/K} \cdot T} \right)^{-1} \gamma (T) = \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T_0} \right) \cdot \tanh \left(\frac{0.063\times 10^{6} \,\text{eV}}{2 \times 8.6173\times 10^{-5} \,\text{eV/K} \cdot T} \right)^{-1} \end{equation} ``` The value of the reference temperature $T_0$ is not entirely clear as it is never stated explicitly, a value of The value of the reference temperature $`T_0`$ is not entirely clear as it is never stated explicitly, a value of $`T_0 = 300 \,\text{K}`$ is assumed. The other parameter values implemented in Allpix Squared are taken from the abstract of \[[@overstraeten]\] as: ```math \begin{equation*} \begin{split} E_0 &= 4.0\times 10^{5} \,\textrm{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\textrm{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\textrm{V/cm}\\ E_0 &= 4.0\times 10^{5} \,\text{V/cm}\\ a_{\infty, e} &= 7.03\times 10^{5} \,\text{/cm}\\ b_{e} &= 1.231\times 10^{6} \,\text{V/cm}\\ \end{split} \qquad \begin{split} a_{\infty, h, \textrm{low}} &= 1.582\times 10^{6} \,\textrm{/cm}\\ a_{\infty, h, \textrm{high}} &= 6.71\times 10^{5} \,\textrm{/cm}\\ b_{h, \textrm{low}} &= 2.036\times 10^{6} \,\textrm{V/cm}\\ b_{h, \textrm{high}} &= 1.693\times 10^{6} \,\textrm{V/cm}\\ a_{\infty, h, \text{low}} &= 1.582\times 10^{6} \,\text{/cm}\\ a_{\infty, h, \text{high}} &= 6.71\times 10^{5} \,\text{/cm}\\ b_{h, \text{low}} &= 2.036\times 10^{6} \,\text{V/cm}\\ b_{h, \text{high}} &= 1.693\times 10^{6} \,\text{V/cm}\\ \end{split} \end{equation*} ``` Loading @@ -135,7 +134,7 @@ features a linear dependence on the electric field strength $`E`$. The coefficie \end{equation} ``` The two parameters $a, b$ are temperature dependent and scale with respect to the reference temperature The two parameters $`a, b`$ are temperature dependent and scale with respect to the reference temperature $`T_0 = 300 \,\text{K}`$ as: ```math Loading @@ -152,16 +151,16 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ ```math \begin{equation*} \begin{split} a_{300, e} &= 0.426}{/V}\\ a_{300, e} &= 0.426 \,\text{/V}\\ c_{e} &= 3.05\times 10^{-4}\\ b_{300, e} &= 4.81\times 10^{5} \,\textrm{V/cm}\\ b_{300, e} &= 4.81\times 10^{5} \,\text{V/cm}\\ d_{e} &= 6.86\times 10^{-4}\\ \end{split} \qquad \begin{split} a_{300, h} &= 0.243 \,\textrm{/cm}\\ a_{300, h} &= 0.243 \,\text{/cm}\\ c_{h} &= 5.35\times 10^{-4}\\ b_{300, h} &= 6.53\times 10^{5} \,\textrm{V/cm}\\ b_{300, h} &= 6.53\times 10^{5} \,\text{V/cm}\\ d_{h} &= 5.67\times 10^{-4}\\ \end{split} \end{equation*} Loading @@ -172,7 +171,7 @@ This model can be selected in the configuration file via the parameter `multipli ## Bologna Model The Bologna model \[[@bologna]\] describes impact ionization for experimental data in an electric field range from $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \,\text{\celsius}`$. The impact ionization $`130 \,\text{kV/cm}`$ to $`230 \,\text{kV/cm}`$ and temperatures up to $`400 \,^{\circ}\text{C}`$. The impact ionization coefficient takes a different form than the previous models and is given by ```math Loading Loading @@ -200,33 +199,33 @@ The parameter values implemented in Allpix Squared are taken from Table 1 of \[[ ```math \begin{equation*} \begin{split} a_{0, e} &= 4.3383}{V}\\ a_{1, e} &= -2.42e-12}{V}\\ a_{0, e} &= 4.3383 \,\text{V}\\ a_{1, e} &= -2.42\times 10^{-12} \,\text{V}\\ a_{2, e} &= 4.1233\\ b_{0, e} &= 0.235 \,\textrm{V}\\ b_{0, e} &= 0.235 \,\text{V}\\ b_{1, e} &= 0\\ c_{0, e} &= 1.6831e4}{V/cm}\\ c_{1, e} &= 4.3796}{V/cm}\\ c_{0, e} &= 1.6831\times 10^{4} \,\text{V/cm}\\ c_{1, e} &= 4.3796 \,\text{V/cm}\\ c_{2, e} &= 1\\ c_{3, e} &= 0.13005}{V/cm}\\ d_{0, e} &= 1.2337e6}{V/cm}\\ d_{1, e} &= 1.2039e3}{V/cm}\\ d_{2, e} &= 0.56703}{V/cm}\\ c_{3, e} &= 0.13005 \,\text{V/cm}\\ d_{0, e} &= 1.2337\times 10^{6} \,\text{V/cm}\\ d_{1, e} &= 1.2039\times 10^{3} \,\text{V/cm}\\ d_{2, e} &= 0.56703 \,\text{V/cm}\\ \end{split} \qquad \begin{split} a_{0, h} &= 2.376 \,\textrm{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\textrm{V}\\ a_{0, h} &= 2.376 \,\text{V}\\ a_{1, h} &= 1.033\times 10^{-2} \,\text{V}\\ a_{2, h} &= 1\\ b_{0, h} &= 0.17714 \,\textrm{V}\\ b_{1, h} &= -2.178\times 10^{-3} \,\textrm{/K}\\ b_{0, h} &= 0.17714 \,\text{V}\\ b_{1, h} &= -2.178\times 10^{-3} \,\text{/K}\\ c_{1, h} &= 0\\ c_{1, h} &= 9.47\times 10^{-3} \,\textrm{V/cm}\\ c_{1, h} &= 9.47\times 10^{-3} \,\text{V/cm}\\ c_{2, h} &= 2.4924\\ c_{3, h} &= 0\\ d_{0, h} &= 1.4043\times 10^{6} \,\textrm{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\textrm{V/cm}\\ d_{2, h} &= 1.4829 \,\textrm{V/cm}\\ d_{0, h} &= 1.4043\times 10^{6} \,\text{V/cm}\\ d_{1, h} &= 2.9744\times 10^{3} \,\text{V/cm}\\ d_{2, h} &= 1.4829 \,\text{V/cm}\\ \end{split} \end{equation*} ``` Loading
